How to make variable transparency hole in a plane with a math function using nodes?
Use the Texture Coordinate node instead of Geometry. The Texture Coordinate provides 'Generated' output that varies from 0 to 1 in each dimension over the whole mesh or Object output that gives zero at the centre of the object.

For a "hole" in the centre of your plane you could use the Object Texture Coordinates and determine the distance from the origin using "distance = sqrt(x^2+y^2+z^2) like this :



Use the result to feed into the Mix factor as in your existing example. To make it variable you can add a Multiply node set to "Clamp" and/or other nodes (add, subtract, Max,Min, etc. - depending on the effect you require) just before the Mix node and adjust this to change the properties of the "hole".